\(\int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx\) [136]

Optimal result

Integrand size = 42, antiderivative size = 182 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {14 c^2 (g \cos (e+f x))^{5/2}}{3 a f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {14 c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{f g (a+a \sin (e+f x))^{3/2}} \] Output:

-14/3*c^2*(g*cos(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e 
))^(1/2)-14*c^2*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin(1/2* 
f*x+1/2*e),2^(1/2))/a/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-4*c* 
(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2)/f/g/(a+a*sin(f*x+e))^(3/2)
 

Mathematica [A] (verified)

Time = 4.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.10 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {2 c (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (21 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {\cos (e+f x)} \left (\cos \left (\frac {1}{2} (e+f x)\right ) (12+\cos (e+f x))+(-12+\cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{3 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{3/2}} \] Input:

Integrate[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e 
 + f*x])^(3/2),x]
 

Output:

(2*c*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(21*El 
lipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + Sqrt[Cos[e 
 + f*x]]*(Cos[(e + f*x)/2]*(12 + Cos[e + f*x]) + (-12 + Cos[e + f*x])*Sin[ 
(e + f*x)/2]))*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(3*f*Cos[e + 
f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^ 
(3/2))
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3329, 3042, 3330, 3042, 3321, 3042, 3121, 3042, 3119}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{3/2}}dx\)

\(\Big \downarrow \) 3329

\(\displaystyle -\frac {7 c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3330

\(\displaystyle -\frac {7 c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3321

\(\displaystyle -\frac {7 c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3121

\(\displaystyle -\frac {7 c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {7 c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}-\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {4 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}}-\frac {7 c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{a}\)

Input:

Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x 
])^(3/2),x]
 

Output:

(-4*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(f*g*(a + a*Sin[e + 
 f*x])^(3/2)) - (7*c*((2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e 
 + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[ 
e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - 
c*Sin[e + f*x]])))/a
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

rule 3321

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* 
(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))   Int[(g 
*Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ 
[b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
 

rule 3329

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2 
*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* 
x])^n/(f*g*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) 
 Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( 
n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] & 
& EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In 
tegersQ[2*m, 2*n, 2*p]
 

rule 3330

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(- 
b)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* 
x])^n/(f*g*(m + n + p))), x] + Simp[a*((2*m + p - 1)/(m + n + p))   Int[(g* 
Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - 
 b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] &&  !LtQ[0, n, m] && IntegersQ[2 
*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1396\) vs. \(2(162)=324\).

Time = 13.16 (sec) , antiderivative size = 1397, normalized size of antiderivative = 7.68

method	result	size
default	\(\text {Expression too large to display}\)	\(1397\)

Input:

int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/2),x,m 
ethod=_RETURNVERBOSE)
 

Output:

-1/3/f/(2*2^(1/2)-3)/(1+2^(1/2))*c/g/a*(30*(1+(-1+2*cos(1/2*f*x+1/2*e)^2)* 
2^(1/2)-2*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*ln(2/g^(1/2)*(g^(1/2)*(g*(-1+2*cos 
(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*cos(1/2*f*x+1/2*e)+g^(1 
/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)-2*cos(1 
/2*f*x+1/2*e)*g-g)/(cos(1/2*f*x+1/2*e)+1))+30*(-1+(1-2*cos(1/2*f*x+1/2*e)^ 
2)*2^(1/2)+2*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*ln(4/g^(1/2)*(g^(1/2)*(g*(-1+2* 
cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*cos(1/2*f*x+1/2*e)+g 
^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)-2*co 
s(1/2*f*x+1/2*e)*g-g)/(cos(1/2*f*x+1/2*e)+1))+24*(-2+(1-2*cos(1/2*f*x+1/2* 
e)^2)*2^(1/2)+4*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*ln(4*g^(1/2)*(g*(-1+2*cos(1/ 
2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*2^(1/2)*cos(1/2*f*x+1/2*e) 
+4*2^(1/2)*g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2 
)^(1/2)+8*cos(1/2*f*x+1/2*e)*g)+24*(2+(-1+2*cos(1/2*f*x+1/2*e)^2)*2^(1/2)- 
4*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*arctanh(2^(1/2)*g^(1/2)*cos(1/2*f*x+1/2*e) 
/(cos(1/2*f*x+1/2*e)+1)/(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e) 
+1)^2)^(1/2))+42*(2+(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e)-1)*2^(1/2) 
+2*cos(1/2*f*x+1/2*e)^2+4*cos(1/2*f*x+1/2*e))*(g*(-1+2*cos(1/2*f*x+1/2*e)^ 
2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*EllipticF((1+2^(1/2))*(-csc(1/2*f*x+1/2 
*e)+cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2) 
+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)*g^2*(-2*(2^(1/2)...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.91 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (21 \, \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (-i \, c g \sin \left (f x + e\right ) - i \, c g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (i \, c g \sin \left (f x + e\right ) + i \, c g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (c g \sin \left (f x + e\right ) + 13 \, c g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{3 \, {\left (a^{2} f \sin \left (f x + e\right ) + a^{2} f\right )}} \] Input:

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/ 
2),x, algorithm="fricas")
 

Output:

-2/3*(21*sqrt(1/2)*sqrt(a*c*g)*(-I*c*g*sin(f*x + e) - I*c*g)*weierstrassZe 
ta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21* 
sqrt(1/2)*sqrt(a*c*g)*(I*c*g*sin(f*x + e) + I*c*g)*weierstrassZeta(-4, 0, 
weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + (c*g*sin(f*x 
+ e) + 13*c*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f 
*x + e) + c))/(a^2*f*sin(f*x + e) + a^2*f)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(3/2)/(a+a*sin(f*x+e))** 
(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/ 
2),x, algorithm="maxima")
 

Output:

integrate((g*cos(f*x + e))^(3/2)*(-c*sin(f*x + e) + c)^(3/2)/(a*sin(f*x + 
e) + a)^(3/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/ 
2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:

int(((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2))/(a + a*sin(e + f*x 
))^(3/2),x)
 

Output:

int(((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(3/2))/(a + a*sin(e + f*x 
))^(3/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:

int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/2),x)
 

Output:

(sqrt(g)*sqrt(c)*sqrt(a)*c*g*( - 2*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + 
f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x) - 4*sqrt(sin(e + f*x) + 1)*sqrt( 
 - sin(e + f*x) + 1)*sqrt(cos(e + f*x)) + 6*int((sqrt(sin(e + f*x) + 1)*sq 
rt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x)*sin(e + f*x))/(sin 
(e + f*x)**3 + sin(e + f*x)**2 - sin(e + f*x) - 1),x)*sin(e + f*x)*f + 6*i 
nt((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*co 
s(e + f*x)*sin(e + f*x))/(sin(e + f*x)**3 + sin(e + f*x)**2 - sin(e + f*x) 
 - 1),x)*f - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(co 
s(e + f*x))*sin(e + f*x)**4)/(cos(e + f*x)*sin(e + f*x)**3 + cos(e + f*x)* 
sin(e + f*x)**2 - cos(e + f*x)*sin(e + f*x) - cos(e + f*x)),x)*sin(e + f*x 
)*f - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f 
*x))*sin(e + f*x)**4)/(cos(e + f*x)*sin(e + f*x)**3 + cos(e + f*x)*sin(e + 
 f*x)**2 - cos(e + f*x)*sin(e + f*x) - cos(e + f*x)),x)*f - 2*int((sqrt(si 
n(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)* 
*3)/(cos(e + f*x)*sin(e + f*x)**3 + cos(e + f*x)*sin(e + f*x)**2 - cos(e + 
 f*x)*sin(e + f*x) - cos(e + f*x)),x)*sin(e + f*x)*f - 2*int((sqrt(sin(e + 
 f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3)/( 
cos(e + f*x)*sin(e + f*x)**3 + cos(e + f*x)*sin(e + f*x)**2 - cos(e + f*x) 
*sin(e + f*x) - cos(e + f*x)),x)*f + int((sqrt(sin(e + f*x) + 1)*sqrt( - s 
in(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**2)/(cos(e + f*x)*sin(...